DS404
Probability and Statistics: Theory and Implementation
Faculty
Andrey Khokhlov
Chief Researcher, IEPT RAS
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
Machine Learning is a powerful tool that is widely used today and is specified in computer science education, while Statistics and Probability are traditionally studied in mathematical departments. Nevertheless, there is a strong relationship between these theories, and it is desirable to be aware of both general principles and differences in approaches. Indeed, machine learning uses mathematical and/or statistical models to gain an overview of the data to make predictions.
The novel contributions in data mining are mostly informal and usually linked with the Bayesian point of view on statistics. But, probability theory itself is more than just Kolmogorov's axiomatic approach or Bayesian reasoning. For instance, the frequentist approach of R. von Mises coexists with novel contributions from quantum probabilities until now. Thus, the overcoming of the formal separation in theoretical understandings may help to avoid confusion and mistakes.
This course tries to show the existing diversity of approaches while staying within classical Kolmogorov's probability theory. The necessary classical theoretical material would be explained as well. We aim to consider several paradoxical situations that arise in practice.
Most examples would be taken from natural sciences and simple situations in data analysis. The outcome is expected to be the practical training in understanding the popular statistical models and the ability to critically read a professional text.
Learning highlights
- Analytical and Monte-Carlo Methods
- Practical Analysis of Real Data
- Appropriate Modelling
- Detecting Exceptional Behaviour
Course outline
15 classes
Session 1
Naive finite models and the need for rigid theory. Paradoxes and Natural Sciences Probabilities in discrete sample spaces.
Session 2
Combinatorics, cases of distinguishable and indistinguishable objects. Generating functions and other computational tools.
Session 3
Conditional probabilities, the independence of events and their formal properties. Bayesian approach for finite and infinite discrete cases
Session 4
Heuristic non-finite models derived by means of the symmetry arguments. Algebra of events and the corresponding mathematical theory.
Session 5
Random variables from formal and heuristic points of view. Examples. Scalar and random vector variables and their properties. Distribution functions.
Session 6
Mutual and group independence of random variables. Models in multidimensional space, sequences and their limits. Probability density functions.
Session 7
Moments and other characteristics of the random variables. Important inequalities. Whether the moments are always defined, whether moments define the distribution.
Session 8
The Law of Large Numbers and what Statistics can do. Sample space and several approaches in Statistics: parametric and non-parametric cases.
Session 9
Integral-valued random variables and their generating functions. Computational techniques, analytical formulas and computer modelling.
Session 10
Basic discrete models and the corresponding discrete random variables. Sequences of random variables and Random Walks model.
Session 11
Asymptotic behaviour of some distributions. The idea of the Central Limit Theorem and its meaning for the Natural Sciences. An experimental illustration.
Session 12
Distribution functions and the general classification of random variables. Back to Statistics: the main problem of Classical statistics. Non-parametric criteria: example.
Session 13
In-depth: Lebeg integration and computational formulas in general cases. An example of a sequence of independent random variables. Kolmogorov’s axioms and other approaches: the model of von Mises and the idea of the quantum probability model.
Session 14
In-depth: Sequences of random variables and several approaches to their limiting behaviour. Characteristic functions. Natural phenomena and mixtures of random variables.
Session 15
Central Limit Theorem and its constraints. Gaussian and non-gaussian statistics. Real data and corresponding traditional assumptions. Simulations.
Course materials
Books
Prerequisites
A standard undergraduate course in calculus is required: geometry, analysis of the real-valued functions (including those of several variables), real and complex linear algebra (matrices eigenvalues and eigenvectors) and at least basic information about the Fourier transform.
Test your skills in solving standard necessary tasks using https://drive.google.com/open?id=1NN72jLCETTRLGznXyantZtRL-o7S3Tcl
We take into account that most undergraduate math programmes include elements of probability theory, therefore our course is not so elementary or narrow.
Some basic experience in MATLAB or PYTHON programming would be appreciated. As an option, all the exercises and the numerical tests can be solved using OCTAVE --- the freeware clone of the MATLAB.
Methodology
We follow modern practice of teaching probabilistic methods both advanced and from scratch: with the deconstruction of standard routine examples along with unexpected counterexamples. Each lesson is divided into a discussion of theory and joint analysis of a computational example or homework. In the middle of the course, an intermediate test is held, which consists of solving typical problems on the basic concepts of the theory.
Grading
After getting his Ph.D. in Algebraic Topology in 1983 Andrey worked in several scientific and/or teaching organisations, among them are the Russian Academy of Sciences, Moscow State University, and Baumann Technology University. The Scientific advising of the graduate and thesis students was part of his activities, not only in Russia, but also in France.
Andrey’s main results in science are linked with geophysical data processing, so naturally his teaching interests are now concentrated in the applied methods of Statistics and their algorithmic implementations. He currently helps his students avoid some common errors within the probabilistic inferences and support their attempts to study Probability and Statistics theory in general.
See full profileApply for this course
Probability and Statistics: Theory and Implementation
by Andrey Khokhlov
Total hours
45 Hours
Dates
Nov 11 - Nov 29, 2024
Fee for single course
€1500
Fee for degree students
€750
How to secure your spot
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FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.